Riesz's theorem for Lumer's Hardy spaces
M Marković - The American Mathematical Monthly, 2020 - Taylor & Francis
The American Mathematical Monthly, 2020•Taylor & Francis
In this note, we obtain a version of the well-known Riesz's theorem on conjugate harmonic
functions for Lumer's Hardy spaces (L h) 2 (Ω) on arbitrary domains Ω: If a real-valued
harmonic function U∈(L h) 2 (Ω) has a harmonic conjugate V on Ω (ie, a real-valued
harmonic function such that U+ iV is analytic on Ω), then U+ iV also belongs to (L h) 2 (Ω),
and for the normalized conjugate we have the norm estimate|| U+ i V||(L h) 2 (Ω)≤ 2|| U||(L
h) 2 (Ω), with the best possible constant.
functions for Lumer's Hardy spaces (L h) 2 (Ω) on arbitrary domains Ω: If a real-valued
harmonic function U∈(L h) 2 (Ω) has a harmonic conjugate V on Ω (ie, a real-valued
harmonic function such that U+ iV is analytic on Ω), then U+ iV also belongs to (L h) 2 (Ω),
and for the normalized conjugate we have the norm estimate|| U+ i V||(L h) 2 (Ω)≤ 2|| U||(L
h) 2 (Ω), with the best possible constant.
Abstract
In this note, we obtain a version of the well-known Riesz’s theorem on conjugate harmonic functions for Lumer’s Hardy spaces on arbitrary domains Ω: If a real-valued harmonic function has a harmonic conjugate V on Ω (i.e., a real-valued harmonic function such that U + iV is analytic on Ω), then U + iV also belongs to , and for the normalized conjugate we have the norm estimate , with the best possible constant.
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